Estimates for -widths of sets of smooth functions on the torus

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• 作者：A. Kushpel^a ; ^{ak357@le.ac.uk" class="auth_mail} ; ^{akshpl@netscape.net" class="auth_mail} ; R.L.B. Stabile^b ; ^{registabile@gmail.com" class="auth_mail} ; ^{ra069475@ime.unicamp.br" class="auth_mail} ; S.A. Tozoni^b ; ^{tozoni@ime.unicamp.br" class="auth_mail}
• 关键词：Torus ; Width ; Multiplier ; Smooth function
• 刊名：Journal of Approximation Theory
• 出版年：July 2014
• 年：2014
• 卷：183
• 期：Complete
• 页码：45-71
• 全文大小：320 K

文摘

In this paper, we investigate n$n$-widths of multiplier operators 52cf0156a6046e0c89c0b715b6b5d3f9">$螞={\left\{{位}_k\right\}}_{k\in {\mathbb{Z}}^d}$ and ${螞}_{\ast }={\left\{{位}_k^{\ast }\right\}}_{k\in {\mathbb{Z}}^d}$, 螞,螞_∗:L^p(T^d)→L^q(T^d)$螞,{螞}_{\ast }:L^p\left({\mathbb{T}}^d\right)\to L^q\left({\mathbb{T}}^d\right)$ on the d$d$-dimensional torus c40bc8f3b00e9dcc273f00806" title="Click to view the MathML source">T^d${\mathbb{T}}^d$, where ${位}_k=位\left(\left|k\right|\right)$ and ${位}_k^{\ast }=位\left({\left|k\right|}_{\ast }\right)$ for a function 位$位$ defined on the interval [0,∞)$[0,\infty )$, with and ${\left|k\right|}_{\ast }={max}_{1\le j\le d}\left|k_j\right|$. In the first part, upper and lower bounds are established for n$n$-widths of general multiplier operators. In the second part, we apply these results to the specific multiplier operators 2c4be506c669839f62c0639bfab">${螞}^{\left(1\right)}={\left\{{\left|k\right|}^{-纬}{(ln\left|k\right|)}^{-尉}\right\}}_{k\in {\mathbb{Z}}^d}$, ${螞}_{\ast }^{\left(1\right)}={\left\{{\left|k\right|}_{\ast }^{-纬}{(ln{\left|k\right|}_{\ast })}^{-尉}\right\}}_{k\in {\mathbb{Z}}^d}$, ${螞}^{\left(2\right)}={\left\{e^{-纬{\left|k\right|}^r}\right\}}_{k\in {\mathbb{Z}}^d}$ and ${螞}_{\ast }^{\left(2\right)}={\left\{e^{-纬{\left|k\right|}_{\ast }^r}\right\}}_{k\in {\mathbb{Z}}^d}$ for 纬,r>0$纬,r>0$ and 尉≥0$尉\ge 0$. We have that 螞⁽¹⁾U_p${螞}^{\left(1\right)}{U}_p$ and ${螞}_{\ast }^{\left(1\right)}{U}_p$ are sets of finitely differentiable functions on T^d${\mathbb{T}}^d$, in particular, 螞⁽¹⁾U_p${螞}^{\left(1\right)}{U}_p$ and ${螞}_{\ast }^{\left(1\right)}{U}_p$ are Sobolev-type classes if 尉=0$尉=0$, and 螞⁽²⁾U_p${螞}^{\left(2\right)}{U}_p$ and ${螞}_{\ast }^{\left(2\right)}{U}_p$ are sets of infinitely differentiable (0<r<1$0<r<1$) or analytic (r=1$r=1$) or entire (r>1$r>1$) functions on T^d${\mathbb{T}}^d$, where U_p$U_p$ denotes the closed unit ball of L^p(T^d)$L^p\left({\mathbb{T}}^d\right)$. In particular, we prove that, the estimates for the Kolmogorov n$n$-widths d_n(螞⁽¹⁾U_p,L^q(T^d))$d_n({螞}^{\left(1\right)}{U}_p,L^q\left({\mathbb{T}}^d\right))$, $d_n({螞}_{\ast }^{\left(1\right)}{U}_p,L^q\left({\mathbb{T}}^d\right))$, d_n(螞⁽²⁾U_p,L^q(T^d))$d_n({螞}^{\left(2\right)}{U}_p,L^q\left({\mathbb{T}}^d\right))$ and $d_n({螞}_{\ast }^{\left(2\right)}{U}_p,L^q\left({\mathbb{T}}^d\right))$ are order sharp in various important situations.